# Lebesgue measure of sets in $\mathbb{R}^N$

Let $$\Omega\subseteq \mathbb{R}^N$$ be an open, bounded and connected set (it can be assumed with smooth boundary if necessary).

Consider $$\phi:\Omega\to\mathbb{R}$$, $$\phi\in C^1(\overline{\Omega})$$ (the Banach space of continuous functions on the closure of $$\Omega$$ having continuous partial derivatives on the closure of $$\Omega$$) satisfying:

$$\bullet\ \Vert \phi\Vert_{C^1(\overline{\Omega})}\leq 1$$

$$\bullet\ \phi^{-1}(0)=\{x\in \Omega\ |\ \phi(x)=0\}\neq \emptyset$$

$$\bullet\ \nabla\phi(x)\neq 0$$ for all $$x\in\phi^{-1}(0)$$.

Is it true that for each $$\varepsilon>0$$ there is a constant $$\delta=\delta(\varepsilon)>0$$ (depending just on $$\varepsilon$$) such that:

$$\lambda (\phi^{-1}(-\delta,\delta))<\varepsilon$$

?

I denote with $$\lambda$$ the Lebesgue measure on $$\mathbb{R}^N$$.

P.S. Our assumption guarantee that any level set $$\phi=c$$ has a null Lebesgue measure.

• What exactly is $C^1(\bar{\Omega})$? In general the idea is that locally you may change the coordinates to $(x_1,\ldots,x_{i-1},\phi(x),x_{i+1},\ldots,x_n)$ by the implicit function theorem. Feb 27, 2021 at 12:37
• I mean that $\phi$ has contiunous partial derivatives on the closure of $\Omega$. Feb 27, 2021 at 12:40

The answer is no even for $$N = 1$$ and $$\Omega = (-1,1)$$, thus $$\bar \Omega = [-1,1]$$. Let $$\phi_1 \colon [-1,1] \to [-1,1]$$ be any increasing function (which may even be in $$C^\infty$$) with $$\phi_1(0) = 0$$ and $$\phi_1'(x) > 0$$ for all $$x \in \bar \Omega$$. Let $$\phi_t := t \cdot \phi_1$$. Then given $$\epsilon > 0$$ and $$C > 0$$ any fixed constant we get $$\lambda(\phi_t^{-1}((-\epsilon,\epsilon)) = 2$$ for $$t < \epsilon$$. Note that the assumptions are fulfilled for any $$t \in (0,1)$$.
• Indeed. I forgot to mention that $\phi$ must be see as a function on $\Omega$ that have an extension to $\overline{\Omega}$. Can you provide an example with $\phi^{-1}(0)\cap \Omega\neq \emptyset$? Feb 27, 2021 at 13:23
• Of course, simply let $\Omega = (-1,1)$ and $\phi_1 \colon [-1,1] \to [-1,1]$ any $C^\infty$-function with $\phi_1(0) = 0$ and $\phi_1'(x) > 0$ for all $x \in \bar \Omega$. I've edited the answer. Feb 27, 2021 at 13:29